Sunday, 18 January 2015

Logical paradoxes

I'm talking about logical paradoxes (This lecture will surprise you: when logic is illogical) at Gresham College on Monday 19th January, which is tomorrow as I write this on Sunday afternoon.  I've been fascinated by these for years, thanks to writers like Martin Gardner, Raymond Smullyan, Douglas Hofstadter.

It's nice to prove things: suppose I want to prove something which is slightly doubtful (like "Arsenal will beat Manchester City this afternoon" - a very unlikely proposition).  Here's a proof from Martin Gardner.  Consider these two statements:

A: Both these statements are false
B: Arsenal will beat Manchester City this afternoon.

Clearly A cannot be true, since if it were it would contradict itself.  So A is false, and if B were also false, then A would be true,  So B must be true.

One proof isn't always enough,  So here's another - this one is Curry's Paradox.  Consider the statement:

If this statement is true, then Arsenal will beat Manchester City this afternoon.

Is this statement true?  It's of the form "If A, then B", and we test that by seeing what happens when A is true.  So assume that the first part of the statement above is true - which means that the whole statement is true, because that is what that clause asserts.  And if that whole statement is true, and the first part is true, then the second part is true. So we have established the truth of the statement above, And if it is true, then Arsenal will win.

So I've proved in two different ways that Arsenal will win, despite almost all the pundits and 76% of the BBC poll thinking the opposite.

ADDED AT 6pm: Arsenal did win.  Which proves the power of mathematical logic.